expression that describes these corrections with the accuracy of the order of 4. frame usually connected with extragalactic objects.

Transcription

1 RUSSIAN JOURNAL OF EARTH SCIENCES, English Translation, VOL, NO, DECEMBER 998 Russian Edition: JULY 998 On the eects of the inertia ellipsoid triaxiality in the theory of nutation S. M. Molodensky Joint Institute of the Physics of the Earth, Bolshaja Gruzinskaja, Moscow 38 Abstract. Modern nutation theories of the Earth planets use the assumption, that the planet s inertia ellipsoid is symmetrical about the axes of rotation. Below we estimate the eects of the inertia ellipsoid triaxiality accurate to ((B ; A)=(C ; A)) 4, A B C are equatorial polar principal moments of inertia, respectively.. Introduction As known, the nutational motion of the Earth planets in space is fully determined by Euler angles # E E E, which describe the position of the body- xed (Tesser's) axes (x y z) with respect to an immobile reference frame. At the same time, dynamic equations of motion of an actual planet model dene only the components (! x,! y,! z ) of the angular velocity vector! with respect to a moving reference frame. In the case of an axially symmetrical planet (equatorial moments of inertia satisfy the condition A = B) the Euler angles are connected with these components through well-known Poinsot's formulae (see, for example, [Lau Lifshitz, 964]), but in the case A 6= B Euler's relations between # E E E (! x! y! z ) present a rather complicated (nonlinear) system of ordinary differential equations which has not been investigated in all details as yet. As was shown by Zharkov et al. [996], in the case of small violation of the axial symmetry (if the parameter =(A ; B)=(C ; A) satises the condition j j) Euler's equations can be solved by using the perturbation method in powers of this parameter in such an approach, rst-order (in ) corrections describe only short-period (semidiurnal) perturbations of the values # E, E, E. Nevertheless, this result cannot be considered as a nal solution of the problem under consideration, because second-order corrections of the order of can contain also longperiod terms, which perturb not only instant values of the nutational amplitudes, but also their mean values. c998 Russian Journal of Earth Sciences. Paper No. TJE983. Online version of this paper was published on July 998. URL: Modern VLBI measurements are most sensible just to these terms, their analysis is most interesting for any practical purposes. Below we present a simple analytical expression that describes these corrections with the accuracy of the order of 4.. Statement of the Problem As mentioned above, modern astrometric measurements determine the motion of xed points of a planetary surface R with respect to an immobile reference frame usually connected with extragalactic objects. At the same time, well-known Liouville's equation M M = L () (! is the angular velocity of the Tesser's reference frame (x y z), M is the angular momentum of the planet, L is the torque of external forces, the dot above symbol denotes the time derivative) determines only the motion of the vector with respect to the moving system of coordinates (x y z). To express the components of the vector R in terms of the known values (! x! y! z ), it is convenient to use the well-known Poinsot's kinematic relations (see, for example, [Zharkov et al., 996]) which connect the trajectory of the vector! with respect to the moving reference frame (x y z) (polhode) the trajectory of the same vector in space (herpolhode). In the case of an axially symmetrical rigid planet ( = the products of inertia I ik vanish), the components M! are connected by the well-known relation M x = A! x M y = B! y M z = C! z () the torque L may be represented in the form L = l(i sin(t) +j cos(t)) (3) 5

2 5 molodensky: effects of the inertia ellipsoid triaxiality i, j are unit vectors oriented along the axes x y, respectively, l =(C ; A)v t =a (4) v t is the amplitude of the nearly diurnal tidal potential, V i = v t a (xz cos(t)+yz sin(t)) = v t a r sin # cos # cos(t ; ) (5) a is the mean radius of the planet, (r # ) are spherical coordinates, is tidal frequency. In the most general case, Liouville equation () represents a system of three nonlinear ordinary dierential equations with respect to three unknown functions! x (t)! y (t)! z (t). In the case =, it has a very simple exact solution. In fact, substituting ()-(4) into (), we obtain! =!(i" cos(t) +j" sin(t) +k) (6) k is a unit vector that coincides with the z axis " = v t(a ; C)(A ;!(A ; C)) a A! +(A ; C)! (7) is the amplitude of nutational motion with respect to the moving reference frame (x y z). The fact that the amplitudes the x y components of vector! in (6) are equivalent is a direct consequence of the axial symmetry of the problem under consideration. In view of this, the surfaces of polhode herpolhode are exactly conical in accordance with Poinsot's kinematic relations, the ratio of apex angles of these cones is dened by the relation = arcsin(;"=( )) (8) the amplitude of nutational motion of a vector R in the body-xed reference frame (x y z) is equal to + " in the case of prograde nutational motion (if the directions of the! motion nutation coincide) to in the opposite case of retrograde motion. At 6= the solutions of dynamic kinematic nonlinear equations can be rather complicated. Because long-period perturbations are most important in practical applications, below we use the method of its approximate solution based on the following procedure:. Dynamic equations () provide expressions for the components! x (t),! y (t),! z (t) of the vector! with respect to the moving reference frame, in which the eects of the inertia ellipsoid triaxiality are taken into account. Inverting Euler's kinematic relations between the known components! x (t)! y (t)! z (t) of the vector! in the moving reference frame, the components! (t),! (t),! 3 (t) of the same vector in space, Euler's angles # E E E, these angles can be expressed in terms of! x (t)! y (t)! z (t). Taking into account, that the kinematic relations are nonlinear, we use the rst-order method of perturbations with respect to the small parameter. 3. To connect short-period rst-order perturbations of the functions! (t)! (t)! 3 (t) long-period second-order perturbations of the same functions, we use the well-known Poinsot's theorem stating that the rotation of an arbitrary triaxial rigid body may be represented as the rolling of the polhode over the herpolhode without sliding. In accordance with this theorem, the ratio of trajectory lengths of the vector! in the moving xed reference frames L L is equal to the ratio of the corresponding periods. The perturbation of the trajectory L in turn consists of () the known terms which describe semidiurnal oscillations () unknown terms of higher orders of smallness which describe the mean radius of this trajectory. As is shown below, this permits the mean radius of the trajectory L to be found with sucient accuracy (of the order of 4 ). 3. Tidal Torque for a Triaxial Body Using expression (5) for the components of tidal potential in Cartesian coordinates, it is easy to nd the tidal torques which act on an axially nonsymmetrical planet L = Z r rv t d = = t [i(c ; B) sin(t) +j(a ; C) cos(t)] a (9) is the total volume of the Earth. Replacing the relation between! y M y in equations () by the formula M y = B! y then substituting () (9) in Liouville's equation (), we obtain A _! x +(C ; B)! y! z = t (C ; B) sin(t) a (a) B _! y +(A ; C)! x! z = t a (A ; C) cos(t) C _! z +(B ; A)! x! y =: (b) (c) Using relations (6) (7), it is easy to see that the

3 molodensky: effects of the inertia ellipsoid triaxiality 53 ratio of terms C _! z (B ; A)! x! y in (c) is of the order of ev t =(ga), e is the attening of the planet g is the gravity on its surface. This ratio is negligibly small for all planets of the solar system, the approximation! z =! = const is valid with a very high accuracy. In this approximation, the solution of equations (a) (b) describing forced nutation precession is # (if this amplitude satises the condition j# j) 4. First-order Approximation for a Slightly Triaxial Body (3d)! =!(i" cos(t) +j" sin(t) +k) (a) " = v t (B ; C)(B ;!(A ; C)) a AB! +(A ; C)(B ; C)! 3 (b) To estimate the eects of triaxiality of the planet in the rst approximation, we rst perform some simple transformations. Multiplying (a) (b) by cos E sin E then subtracting the results, we express # E in terms of " " as follows: _# E =!(" cos E ; " sin E ): (4a) " = v t (A ; C)(A ;!(B ; C)) a AB! +(A ; C)(B ; C)! 3 (c) Analogously, multiplying (a) (b) by sin E cos E then summing the results, we have Substituting relations () in the well-known Euler relations between Euler angles (# E E E ) (! x! y! z )[Lau Lifshitz, 964]! x = _ E sin # E sin E + _ # E cos E (a)! y = _ E sin # E cos E + _ # E sin E (b)! z = _ E cos # E cos E + _ E (c) we obtain a system of three nonlinear ordinary dierential equations with respect to three unknown functions # E E E. In the case of an axially symmetrical body (A = B " = " = ", " is dened by (7)) these equations have an elementary (Poinsot) solution in the form _ E sin # E =!(" sin E + " cos E ) (4b) To linearize these equations, their solutions are represented by the superposition E = + E = + # E = # + # (5) # are the solutions of axially symmetrical problem (3a)-(3c), # are small corrections due to the triaxiality eects of the inertia ellipsoid. Substituting (5) (a) into (4a), (4b), (c) neglecting the terms of the order of,we obtain _ sin # + _ sin # + # _ cos # =!" cos(t)(sin + cos )+!" sin(t)(cos ; sin ) (6a) # = arctan (3a) _# + _ # =!" cos(t)(cos ; sin );!" sin(t)(sin + cos ) (6b) = ; t (3b) = sin # t + const (3c) which show, that the angular frequency amplitude of nutational motion in space are _ = sin # _ cos # + _ cos # ; # _ sin # + + _ + _ =: (6c) Transforming, in accordance with (3b), the main terms in the right-h sides of these equations,! " + "!" cos(t) sin " sin(t) cos = sin( + t) " ; "! " + " " ; " sin( ; t) = cos(t)!" cos(t)cos ;!" sin(t) sin =

4 54 molodensky: effects of the inertia ellipsoid triaxiality! " + " cos( + t) " ; "! " ; " sin(t) cos( ; t) = taking into account, that the zero-order values of # are connected with "! by relations (3), we obtain, after dierentiation of these relations with respect to the time, Since the system of equations (9) is of the third order, it has three linearly independent homogeneous solutions. It is easy to show, however, that these solutions are not interesting for the problem under consideration, because they describe the small variations in Euler angles caused by an arbitrary innitesimal rotation of the xed reference frame. The inhomogeneous solution of (9) has the form _ sin # =!" =! " + " (7a) = ; = a sin(t) # = b cos(t) () _# = (7b) _ cos # + _ =: (7c) Using these relations, equations (6) can be rewritten as follows: _ sin # + # _ cos # ;! " + " =! " ; " cos(t) sin(t) = (8a) a = ; (" ; " )(3 )( )) "(! ; 3 ) b = ;a : _# +! " + " =! " ; " _ cos # ; # _ sin # + _ : sin(t) (8b) (8c) Eliminating # _ from these relations with the help of (3a) (7c), we nally obtain the system of three inhomogeneous linear ordinary dierential equations with the coecients c _ + c # = cos(t) _# + c 3 = sin(t) _ + c _ 4 ; c 3 # =: c =sin# = (() +( ) ) = c =!" cot # = c 4 =cos# = c 3 =!" (() +( ) ) = =!(" ; " )=: (9) These expressions fully determine rst-order shortperiod perturbations of the nutational precessional motion caused by the triaxiality of the planet s gure. To nd short-period motion perturbations of the vector!, we introduce a xed Cartesian reference frame (e e e 3 )insuchaway that the axis e 3 coincides with the axis of the unperturbed cone of herpolhode, E is equal to the angle between the nodal line vector e. Then the relations between components (!!! 3 ) Euler angles (# E E E ) are completely analogous to (a)-(c) after replacement of E by E E by E :! = _ E sin # E sin E + _ # E cos E (a)! = _ E sin # E cos E ; _ # E sin E (b)! 3 = _ E cos # E sin E + _ E : (c) In the case of nutational motion of any actual planet, its spatial amplitude =( ) does not exceed a few seconds of arc, we can use, instead of (3a), a more simple expression # = : () Substituting (5), (3b), (3c), (), (7) into (), we have

5 molodensky: effects of the inertia ellipsoid triaxiality 55! =(; ; a cos(t))( + +b cos(t)) sin(( )t + a sin(t)); ;b sin(t) cos(( )t + a sin(t))! =(; ; a cos(t))( + +b cos(t)) cos(( )t + a sin(t))+ +b sin(t) cos(( )t + a sin(t)) (3a) (3b) part of this function f (s) (B;A) which is connected with short-period (semidiurnal) perturbations of herpolhode. Using this value, the total perturbation of herpolhode length can be represented as a sum of the known shortperiod perturbation L () f (s) (B ;A) unknown longperiod perturbation L () f () (B ;A) that describes vari- ation in the \mean" herpolhode radius: = L () (s) (f L () f (B ; A) = () (B ; A)+f (B ; A)): (5a)! 3 =!: (3c) Like the rst-order corrections to Euler angles (), these expressions contain only short-period (semidiurnal) perturbations. 5. Long-period terms in the second-order approximation Now, we will use the well-known Poinsot's theorem stating that the rolling of polhode over herpolhode takes place without sliding. On the strength of this theorem, a general method for the calculation of long-period herpolhode perturbations may be formulated as follows. Let the total length of the vector! trajectory in the moving reference frame (x y z) is dened by the expression L = L () ( + f (B ; A)) (4a) L () is the length of the vector in the case of an axially symmetrical planet (A = B) f (B ; A) isa small correction due to +eects of triaxiality. Because this trajectory coincides with circle (6) if A = B with ellipse (a) if A 6= B, the function f actually describes the known dierence between the length of ellipse (a) with the semiaxes!"!" the length!" of circle (6). Likewise, using relations (3), the total length of the! trajectory in the xed reference frame (e e e 3 )can be represented in the form L = L () ( + f (B ; A)) (4b) L () = j=( )jl () is the total length of the herpolhode in the biaxial case f (B ; A) describes the dierence between the lengths in the triaxial biaxial cases. As mentioned before, the absence of exact expressions for the Euler angles in terms of known components! x (t)! y (t) precludes direct determination of this function. However, expressions (3) readily provide the In accordance with Poinsot's theorem, the condition of the polhode rolling over herpolhode without sliding is that the trajectories described by the angular velocity vector during equal time intervals have the same lengths in moving xed reference frames: L =T = L () ( + f (B ; A))=T = = L () ( + f (B ; A))=T = = j=( )jl () (s) ( + f (B ; A)+ +f () (B ; A))=T T T are full revolution periods of the vector! in the moving xed reference frames, respectively. Since these values are connected through the relation T = j=( )jt the unknown value f () (B ; A) can be expressed in terms of the known functions f (B ;A) f (s) (B ;A) in the following, very simple form: f () (B ; A) =f (B ; A) ; f (s) (B ; A) (5b) Then, variation in the mean radius of herpolhode (which coincides with variation in the nutation amplitude A multiplied by!) may be represented in the form A = A f () (B ; A) = (6) = A (f (B ; A) ; f (s) (B ; A)) A is the unperturbed nutation amplitude corresponding to the axially symmetrical approximation. In accordance with (), the complete trajectory length of the vector! in the moving reference frame (x y z) is L =! =! Z Z (" cos + " sin ) = d = ( " + " + " ; " cos()) = d:

6 56 molodensky: effects of the inertia ellipsoid triaxiality Developing the integr function as the Taylor s series in small parameter =(" ; + ) with " )=(" " the help of (b) (c) neglecting terms of the order of 4,wehave Z _! = k cos( + )+k sin( + ) (3a) _! = k sin( + )+k cos( + ) (3b) L =! (( + " " ) = ; ; (" ; " ) 8 p (" + " )3= cos ())d = =!(" + (" ; " ) 8" ; (" ; " ) 64" 3 )= =!(" + (" ; " ) ): 6" (7a) Comparing this expression relation (4a) yields f (A ; B) = (" ; " ) L = lim N! N 6" (7b) the nutation amplitudes " " " are dened by expressions (b), (c), (7), respectively. Strictly speaking, the denition of the complete trajectory length of the vector! in the xed reference frame (e e e 3 needs some renement, because this trajectory is generally nonclosed. In fact, it is obvious that, since the period of short-period perturbations is equal to = the period of nutational motion is equal to =j j, this trajectory is closed only if = k( ), k is any integer number. Of course, nutation frequencies of actual planets do not satisfy this condition, the herpolhode is not a closed curve. In order to apply Poinsot's theorem to the case of nonclosed trajectory of the angular velocity vector in space, we dene the \mean length" of this trajectory as follows: Z N ) ) ) = dt: (8) Substituting (3) into (8) exping, as before, the integr function in Taylor's series with the accuracy of the order of 4,we obtain lim N! N L = j Z j N ( + ; )dt (9a) _! 3 = (3c) k = ; ; (b( ))+ +a ( cos(t); ;(ab( ) +4a ) cos (t) k =(4a +b( )) sin(t) = = ;a"! sin(t): (3a) (3b) Substituting these expressions into (9), (3a), (3b) neglecting the terms of the fourth higher orders, we have (_! ) +(_! ) +(_! 3 ) = k + k = =() ( ; a! ;! + cos(t)+ +a (! ; ) ; 4( ) ( ) cos (t)+ +4a! ( ) sin (t)) j j lim N! N Z N ( + )dt = L () ( + a 5! ; 3 ; 6! 4( ) ) j j lim N! N Z N 8 dt = (3a) (3b) (3c) (! ; L () ) (a 4( ) ) Z N 3 dt =: (33) = () [( _! ) +(_! ) +(_! 3 ) ] ; : Substituting (a-c) into (6a, b), we have (9b) Summing these expressions, we nally obtain L = j=( )jl () (s) ( + f (B ; A)) (34a)

7 molodensky: effects of the inertia ellipsoid triaxiality 57 f (s) (B ; A) =a! ;! ; ( ) = = (" ; " ) (3 ) (! ;! ; ) 4" (! ; 3 ) = = (" ; " ) (! ;! ; ) 4" ( ;!) (34b) Substituting (34b) (6b) into (3), we obtain the nal value of the correction to the mean nutation amplitudes A = A + A = A ( + f () (B ; A)) = = A ( + (" ; " ) 4" ( 4 ; (! ;! ; ) ( ;!) )) = = A ( + (" ; " ) 6" ( )(5 ; 3!) ( ;!) ): (35) respectively, T ==! is one sidereal day). Nutational motion with the period T s = T = takes place only in the case of synchronised diurnal orbital rotation (if at the points of equinoxes the orientation of the planet's principal equatorial moments of inertia is invariable in space). In this case, the nutation amplitudes are obviously determined not by the mean values of A, B averaged over the total period, but by their \eective" values in the vicinity ofpoints at which the tidal torque is maximum. In this specic case, the corrections to the mean nutation amplitudes are proportional to (" " )=" rather than (" " ) =". This qualitative consideration explains, why second-order correction (35) tends to in- nity in the limiting case!!. Acknowledgement. This work was supported by the Russian Foundation for Basic Research, project No Note that this correction vanishes if = ;! or = 3!=5. The rst root corresponds to the precession frequency. Vanishing of this expression for the precession frequency implies that the precessional constant depends only on the value ; (A + B)= but not on the dierence of equatorial moments of inertia B ; A. The second root, as well as the denominator root = ;!, corresponds to the case of high nutation frequencies in space as compared with the period of diurnal rotation (these frequencies correspond to the spatial periods of nutational motion T s =5T =8T s = T =, References Lau, L. D. E. M. Lifshitz, Mechanics, Moscow, Nauka, 964 (in Russian). Molodensky, S.M.,Tides, Nutation, Internal Structure of the Earth, Moscow, Nauka, 98, 4 pp. (in Russian). Zharkov V. N., S. M. Molodensky, E. Groten, A. Brzezinski, P. Varga. The Earth its rotation. Low-frequency geodynamics, Heidelberg, Wichman Verlag, 996, 53 pp. (Received May )